An old theorem of Pospisil asserts that for any infinite set $I$ the power-set algebra $\wp(I)$ has $\exp \exp |I|$ many maximal ideals containing the ideal of finite sets. This result is published in a rather obscure Czech journal but it seems it should be well-known and described in many textbooks/monographs. I would appreciate any references for that.

Also, I am interested in more general results, that is, what are the sufficient conditions for a given Boolean algebra $\mathcal{A}\subseteq \wp(I)$ with $\mbox{fin}(I)\subseteq \mathcal{A}$ to have $\exp \exp |I|$ many maximal ideals containing $\mbox{fin}(I)$.

I am sure this is in Koppelberg's Handbook of Boolean Algebras, though I have not checked (I will check later).
A proof of Pospisil's result is in this survey-paper of mine
that will appear in RIMS Kokyuroku.

I am not sure I completely understand the second part of your question, though.
Are you asking for conditions when you have this many maximal ideals?
For example, it could happen that $\mathcal A$ is only of size $|I|$ and you have at most
$2^{|I|}$ maximal ideals.